An exact integral equation for solitary waves (with new numerical results for some ‘internal’ properties)

A novel exact within potential flow integral equation for the solitary wave profile, ns(x), is presented and numerically solved utilizing a parametric form for the profile and ‘tailored quadrature’ methods. We believe the profiles and properties of such waves, so obtained, to be the most accurate (to date). From the profiles, certain internal properties of interest, such as the shapes and properties of internal streamlines, internal velocities and pressures, that appear to be hitherto unevaluated in the literature, are here presented. In the outskirts, it is shown that the Stokes form for the exponential decay, ns(x) ~ e-μx/h, of the surface profile is also valid for all streamlines. The amplitude of this exponential decay is numerically obtained for all solitary wave surface profiles. The analagous decay amplitude of an internal streamline is shown to be related to the surface profile amplitude via a simple quadrature. Like several other properties of solitary waves, it is found that the surface profile outskirts decay amplitude, as well as the pressure on the canal bed directly underneath the crest, are largest for waves of lesser height than the ‘maximum’ wave.